CapStone Project_2025
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Diabetes mellitus (DM) is a major public health concern closely associated with factors such as obesity, age, race, and gender. Identifying these associated risk factors is essential for targeted interventions D’Angelo and Ran (2025). Logistic Regression (traditional) that estimates the association between risk factors and outcomes is insufficient in analyzing the complex healthcare data (DNA sequences, imaging, patient-reported outcomes, electronic health records (EHRs), longitudinal health measurements, diagnoses, and treatments. Zeger et al. (2020). Classical maximum likelihood estimation (MLE) yields unstable results in samples that are small, have missing data, or presents quasi- and complete separation.
Bayesian hierarchical models using Markov Chain Monte Carlo (MCMC) allow analysis of multivariate longitudinal healthcare data with repeated measures within individuals and individuals nested in a population. By integrating prior knowledge and including exogenous (e.g., age, clinical history) and endogenous (e.g., current treatment) covariates, Bayesian models provide posterior distributions and risk predictions for conditions such as pneumonia, prostate cancer, and mental disorders. Parametric assumptions remain a limitation of these models.
In Bayesian inference Chatzimichail and Hatjimihail (2023), Bayesian inference has shown that parametric models (with fixed parameters) often underperform compared to nonparametric models, which do not assume a prior distribution. Posterior probabilities from Bayesian approaches improve disease classification and better capture heterogeneity in skewed, bimodal, or multimodal data distributions. Bayesian nonparametric models are flexible and robust, integrating multiple diagnostic tests and priors to enhance accuracy and precision, though reliance on prior information and restricted access to resources can limit applicability. Combining Bayesian methods with other statistical or computational approaches helps address systemic biases, incomplete data, and non-representative datasets.
The Bayesian framework described by R. van de Schoot et al. (2021) highlights the role of priors, data modeling, inference, posterior sampling, variational inference, and variable selection.Proper variable selection mitigates multicollinearity, overfitting, and limited sampling, improving predictive performance. Priors can be informative, weakly informative, or diffuse, and can be elicited from expert opinion, generic knowledge, or data-driven methods. Sensitivity analysis evaluates the alignment of priors with likelihoods, while MCMC simulations (e.g., brms, blavaan in R) empirically estimate posterior distributions. Spatial and temporal Bayesian models have applications in large-scale cancer genomics, identifying molecular interactions, mutational signatures, patient stratification, and cancer evolution, though temporal autocorrelation and subjective prior elicitation can be limiting.
Bayesian normal linear regression has been applied in metrology for instrument calibration using conjugate Normal–Inverse-Gamma priors Klauenberg et al. (2015). Hierarchical priors add flexibility by modeling uncertainty across multiple levels, improving robustness and interpretability. Bayesian hierarchical/meta-analytic linear regression incorporates both exchangeable and unexchangeable prior information, addressing multiple testing challenges, small sample sizes, and complex relationships among regression parameters across studies Leeuw and Klugkist (2012)
A sequential clinical reasoning model Liu et al. (2013) Sequential clinical reasoning models demonstrate screening by adding predictors stepwise: (1) demographics, (2) metabolic components, and (3) conventional risk factors, incorporating priors and mimicking clinical evaluation. This approach captures ecological heterogeneity and improves baseline risk estimation, though interactions between predictors and external cross-validation remain limitations.
Bayesian multiple imputation with logistic regression addresses missing data in clinical research Austin et al. (2021) in clinical research by classifying missing values (e.g., patient refusal, loss to follow-up, mechanical errors) as MAR, MNAR, or MCAR. Multiple imputation generates plausible values across datasets and pools results for reliable classification of patient health status and mortality.
The present study aims performs Bayesian logistic regression to predict diabetes status and evaluate the associations between diabetes and predictors (body mass index (BMI), age (≥20 years), gender, and race). The study anakyzes a retrospective dataset (2013–2014 NHANES survey data). It is based on a complex sampling design, characterized by stratification, clustering, and oversampling of specific population subgroups, rather than uniform random sampling. A Bayesian analytical approach addresses challenges posed by dataset anomalies such as missing data, complete case analysis, and separation that limit the efficiency and reliability of traditional logistic regression in predicting health outcomes.
##Logical Flow for Analysis
# loading packages
options(repos = c(CRAN = "https://cloud.r-project.org"))
install.packages("nhanesA")
library ("nhanesA")
library(tidyverse)
library(knitr)
library(ggthemes)
library(ggrepel)
library(dslabs)
library(Hmisc)
library(dplyr)
library(tidyr)
library(forcats)
library(ggplot2)
library(classpackage)
library(janitor)
install.packages("gt")
library(gt)
library(survey)
library(DataExplorer)
library(logistf)DIQ050 a secondary variable describing treatment status (insulin use) to exclude those casesMerged dataset is cleaned and exploratory data analysis conducted to report results and visualizations for 10175 observations and 10 variables where - - race categorized in 5 levels - age range (0-80 years) - gender (male and female) - Diabetes grouped from (DIQ010 and DIQ050) BMI as continuous.
# weighted means of each variable
str(merged_data)'data.frame': 10175 obs. of 10 variables:
$ SEQN : num 73557 73558 73559 73560 73561 ...
$ RIDAGEYR: num 69 54 72 9 73 56 0 61 42 56 ...
$ RIAGENDR: Factor w/ 2 levels "Male","Female": 1 1 1 1 2 1 1 2 1 2 ...
$ RIDRETH1: Factor w/ 5 levels "Mexican American",..: 4 3 3 3 3 1 3 3 2 3 ...
$ SDMVPSU : num 1 1 1 2 2 1 1 1 2 1 ...
$ SDMVSTRA: num 112 108 109 109 116 111 105 114 106 112 ...
$ WTMEC2YR: num 13481 24472 57193 55767 65542 ...
$ BMXBMI : num 26.7 28.6 28.9 17.1 19.7 41.7 NA 35.7 NA 26.5 ...
$ DIQ010 : Factor w/ 5 levels "Yes","No","Borderline",..: 1 1 1 2 2 2 NA 2 2 2 ...
$ DIQ050 : Factor w/ 4 levels "Yes","No","Refused",..: 1 1 1 2 2 2 NA 2 2 2 ...plot_str(merged_data)
head(merged_data) SEQN RIDAGEYR RIAGENDR RIDRETH1 SDMVPSU SDMVSTRA WTMEC2YR BMXBMI
1 73557 69 Male Non-Hispanic Black 1 112 13481.04 26.7
2 73558 54 Male Non-Hispanic White 1 108 24471.77 28.6
3 73559 72 Male Non-Hispanic White 1 109 57193.29 28.9
4 73560 9 Male Non-Hispanic White 2 109 55766.51 17.1
5 73561 73 Female Non-Hispanic White 2 116 65541.87 19.7
6 73562 56 Male Mexican American 1 111 25344.99 41.7
DIQ010 DIQ050
1 Yes Yes
2 Yes Yes
3 Yes Yes
4 No No
5 No No
6 No Nosummary(merged_data) SEQN RIDAGEYR RIAGENDR
Min. :73557 Min. : 0.00 Male :5003
1st Qu.:76100 1st Qu.:10.00 Female:5172
Median :78644 Median :26.00
Mean :78644 Mean :31.48
3rd Qu.:81188 3rd Qu.:52.00
Max. :83731 Max. :80.00
RIDRETH1 SDMVPSU SDMVSTRA
Mexican American :1730 Min. :1.000 Min. :104.0
Other Hispanic : 960 1st Qu.:1.000 1st Qu.:107.0
Non-Hispanic White :3674 Median :1.000 Median :111.0
Non-Hispanic Black :2267 Mean :1.484 Mean :110.9
Other Race - Including Multi-Racial:1544 3rd Qu.:2.000 3rd Qu.:115.0
Max. :2.000 Max. :118.0
WTMEC2YR BMXBMI DIQ010 DIQ050
Min. : 0 Min. :12.10 Yes : 737 Yes : 220
1st Qu.: 12562 1st Qu.:19.70 No :8841 No :9545
Median : 20175 Median :24.70 Borderline: 185 Refused : 1
Mean : 30585 Mean :25.68 Refused : 1 Don't know: 2
3rd Qu.: 36748 3rd Qu.:30.20 Don't know: 5 NA's : 407
Max. :171395 Max. :82.90 NA's : 406
NA's :1120 library(dplyr)
library(knitr)
plot_intro(merged_data, title="Figure 1 (Merged dataset). Structure of variables and missing observations.")
plot_missing(merged_data, title="Figure 2(Merged dataset). Breakdown of missing observations.")
# print(glimpse(merged_data))
print(table(merged_data$BMDBMIC, useNA = "ifany"))< table of extent 0 >print(table(merged_data$DIQ010, useNA = "ifany"))
Yes No Borderline Refused Don't know <NA>
737 8841 185 1 5 406 # ---- Coercion helpers (handle labelled/character) ----
to_num <- function(x) {
if (is.numeric(x)) return(x)
xc <- as.character(x)
n <- suppressWarnings(readr::parse_number(xc))
if (mean(is.na(n)) > 0.80) {
xlow <- tolower(trimws(xc))
n <- dplyr::case_when(
xlow %in% c("1","yes","yes, told") ~ 1,
xlow %in% c("2","no","no, not told") ~ 2,
xlow %in% c("3","borderline") ~ 3,
xlow %in% c("7","refused") ~ 7,
xlow %in% c("9","don't know","dont know","unknown") ~ 9,
TRUE ~ NA_real_
)
}
as.numeric(n)
}
merged_data <- merged_data %>%
mutate(
DIQ010 = to_num(DIQ010),
DIQ050 = to_num(if (!"DIQ050" %in% names(.)) NA_real_ else DIQ050),
BMXBMI = suppressWarnings(as.numeric(BMXBMI)),
RIDAGEYR = suppressWarnings(as.numeric(RIDAGEYR)),
RIAGENDR = suppressWarnings(as.numeric(RIAGENDR)),
RIDRETH1 = suppressWarnings(as.numeric(RIDRETH1)),
SDMVPSU = suppressWarnings(as.numeric(SDMVPSU)),
SDMVSTRA = suppressWarnings(as.numeric(SDMVSTRA)),
WTMEC2YR = suppressWarnings(as.numeric(WTMEC2YR))
)
# ---- Diagnostics BEFORE save ----
cat("DIQ010 counts BEFORE save:\n")DIQ010 counts BEFORE save:print(table(merged_data$DIQ010, useNA = "ifany"))
1 2 3 7 9 <NA>
737 8841 185 1 5 406 cat("Count with DIQ010 in {1,2}:", sum(merged_data$DIQ010 %in% c(1,2), na.rm = TRUE), "\n")Count with DIQ010 in {1,2}: 9578 # ---- Save to file for reuse ----
dir.create("data", showWarnings = FALSE)
# ---- Save ----
dir.create("data", showWarnings = FALSE, recursive = TRUE)
saveRDS(merged_data, "data/merged_2013_2014.rds")
message("Saved: data/merged_2013_2014.rds")We transformed special codes (3,7,) to NA and included all NAs in the analysis. Visualization of missing data is presented below.
A final analytic dataset created (‘adult’) with “NH White” and “Male” as the reference group for analysis
##
# ---------------- Basic Exploration (adults) ----------------
# Keep adults only and build analysis variables
adult <- merged_data %>%
dplyr::filter(RIDAGEYR >= 20) %>%
dplyr::transmute(
# --- keep survey design variables so svydesign() can see them ---
SDMVPSU, SDMVSTRA, WTMEC2YR,
# --- outcome: DIQ010 (1 yes, 2 no; 3/7/9 -> NA) ---
diabetes_dx = dplyr::case_when(
DIQ010 == 1 ~ 1,
DIQ010 == 2 ~ 0,
DIQ010 %in% c(3, 7, 9) ~ NA_real_,
TRUE ~ NA_real_
),
# --- predictors (raw) ---
bmi = BMXBMI,
age = RIDAGEYR,
# sex (1=Male, 2=Female)
sex = forcats::fct_recode(factor(RIAGENDR), Male = "1", Female = "2"),
# race (5-level)
race = forcats::fct_recode(
factor(RIDRETH1),
"Mexican American" = "1",
"Other Hispanic" = "2",
"NH White" = "3",
"NH Black" = "4",
"Other/Multi" = "5"
),
# keep DIQ050 so we can safely reference it (may be absent/NA in some rows)
DIQ050 = DIQ050
) %>%
# standardize continuous predictors
dplyr::mutate(
age_c = as.numeric(scale(age)),
bmi_c = as.numeric(scale(bmi)),
bmi_cat = cut(
bmi,
breaks = c(-Inf, 18.5, 25, 30, 35, 40, Inf),
labels = c("<18.5","18.5–<25","25–<30","30–<35","35–<40","≥40"),
right = FALSE
)
) %>%
# adjust outcome: if female & DIQ050==1 ("only when pregnant"), set to 0 (not diabetes)
dplyr::mutate(
diabetes_dx = ifelse(sex == "Female" & !is.na(DIQ050) & DIQ050 == 1, 0, diabetes_dx)
)
# Make NH White the reference level for race (clearer interpretation)
adult <- adult %>%
dplyr::mutate(
race = forcats::fct_relevel(race, "NH White")
)
# --- sanity checks ---
cat("Adults n =", nrow(adult), "\n")Adults n = 5769 glimpse(adult)Rows: 5,769
Columns: 12
$ SDMVPSU <dbl> 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2…
$ SDMVSTRA <dbl> 112, 108, 109, 116, 111, 114, 106, 112, 112, 113, 116, 114…
$ WTMEC2YR <dbl> 13481.04, 24471.77, 57193.29, 65541.87, 25344.99, 61758.65…
$ diabetes_dx <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ bmi <dbl> 26.7, 28.6, 28.9, 19.7, 41.7, 35.7, NA, 26.5, 22.0, 20.3, …
$ age <dbl> 69, 54, 72, 73, 56, 61, 42, 56, 65, 26, 76, 33, 32, 38, 50…
$ sex <fct> Male, Male, Male, Female, Male, Female, Male, Female, Male…
$ race <fct> NH Black, NH White, NH White, NH White, Mexican American, …
$ DIQ050 <dbl> 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2…
$ age_c <dbl> 1.13241831, 0.27835981, 1.30323001, 1.36016725, 0.39223428…
$ bmi_c <dbl> -0.33588609, -0.07028101, -0.02834336, -1.31443114, 1.7609…
$ bmi_cat <fct> 25–<30, 25–<30, 25–<30, 18.5–<25, ≥40, 35–<40, NA, 25–<30,…##Exploratory Description and Visualization of Adult dataset (> 20 years)**
Observations - 5769 observations Survey design: SDMVPSU, SDMVSTRA, WTMEC2YR Outcome: diabetes_dx (numeric 0/1) Covariates: bmi, age, sex, race, DIQ050 Centered covariates: age_c, bmi_c BMI categories: bmi_cat
NHANES is a national surveys based on complex sampling designs (oversampling certain groups (e.g., minorities, older adults) to ensure representation. They use multistage sampling to represent the U.S. population, so we apply sampling weights, strata, and PSU (primary sampling units) for valid estimates.
We use survey design in regression anlaysis to avoid - - bias prevalence estimates (e.g., mean BMI or diabetes %) - underestimation of standard errors - incorrect inference for population-level parameters.
# data exploration
print(table(adult$diabetes_dx, useNA = "ifany"))
0 1 <NA>
4974 618 177 print(table(adult$sex, useNA = "ifany"))
Male Female
2758 3011 print(table(adult$race, useNA = "ifany"))
NH White Mexican American Other Hispanic NH Black
2472 767 508 1177
Other/Multi
845 if (sum(!is.na(adult$diabetes_dx)) == 0) {
stop("Too few non-missing outcomes for modeling (n = 0). Check DIQ010 upstream.")
}
# (optional plots omitted for brevity)
# save for downstream
if (!dir.exists("data")) dir.create("data", recursive = TRUE)
saveRDS(adult, "data/adult_cleaned_2013_2014.rds")ggplot(adult, aes(x = age)) +
geom_histogram(binwidth = 5, fill = "skyblue", color = "white") +
labs(
title = "Distribution of Age >20 years",
x = "Age (years)",
y = "Count"
) +
theme_minimal()
ggplot(adult, aes(factor(diabetes_dx))) +
geom_bar(fill = "steelblue") +
labs(title="Diabetes Outcome Distribution in >20 years age group", x="diabetes_dx (0=No, 1=Yes)", y="Count")
ggplot(adult, aes(factor(bmi_cat))) +
geom_bar(fill = "steelblue") +
labs(title="Diabetes Outcome Distribution by BMI in >20 years age group", x="bmi_cat")
ggplot(adult, aes(x = factor(diabetes_dx), y = bmi)) +
geom_boxplot(fill = "skyblue") +
labs(
title = "BMI Distribution by Diabetes Diagnosis in >20 years age group",
x = "Diabetes Diagnosis (0 = No, 1 = Yes)",
y = "BMI"
) +
theme_minimal()
# plots for adult data bmi categories and race categories
ggplot(adult, aes(x = factor(race), fill = factor(diabetes_dx))) +
geom_bar(position = "dodge") +
labs(
title = "Diabetes Diagnosis by Race in >20 years age group",
x = "Race/Ethnicity",
y = "Count",
fill = "Diabetes Diagnosis\n(0 = No, 1 = Yes)"
) +
theme_minimal() +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
ggplot(adult, aes(x = factor(bmi_cat), fill = factor(diabetes_dx))) +
geom_bar(position = "dodge") +
labs(
title = "Diabetes Diagnosis by BMI in >20 years age group",
x = "BMI",
y = "Count",
fill = "Diabetes Diagnosis\n(0 = No, 1 = Yes)"
) +
theme_minimal() +
theme(axis.text.x = element_text(angle = 45, hjust = 1))
We conducted frequentist method Multiple Logistic regression on a survey-weighted dataset, for complete case analysis and data exploration
DIQ010).Summary of Bayesian regression presented under following headings - Posterior Predictive Probabilities - Posterior Mean, Median, credible Intervals - Posterior Probability (Outcome=1) - Comparison with External Prevalence (population prevalence) - Posterior Model Fit Metrics - Prior versus Posterior Coefficient Distributions - Posterior Predictive Checks - Uncertainty Quantification
Bayesian Logistic Regression model
\[ \text{logit}(P(Y_i=1)) = \beta_0 + \beta_1 \cdot Age_i + \beta_2 \cdot BMI_i + \beta_3 \cdot Race_i + \beta_4 \cdot Gender_i \] Linear Regression equation:
\[ y_i = \beta_0 + \beta_1 X_1 +\varepsilon_i \] Diagnostics for Bayesian Logictic Regression is presented under headings
# Modeling
library(broom)
library(mice)
library(brms)
library(posterior)
library(bayesplot)
library(knitr)
# --- Guardrails for modeling ---
n_outcome <- sum(!is.na(adult$diabetes_dx))
if (n_outcome == 0) stop("Too few non-missing outcomes for modeling. n = 0")
# Ensure factors and >=2 observed levels among complete outcomes
adult <- adult %>%
dplyr::mutate(
sex = if (!is.factor(sex)) factor(sex) else sex,
race = if (!is.factor(race)) factor(race) else race
)
if (nlevels(droplevels(adult$sex[!is.na(adult$diabetes_dx)])) < 2)
stop("sex has <2 observed levels after filtering; check data availability.")
if (nlevels(droplevels(adult$race[!is.na(adult$diabetes_dx)])) < 2)
stop("race has <2 observed levels after filtering; check Data Prep.")
# Survey-weighted complete-case
# Build a logical filter on the original adult data (same length as design$data)
keep_cc <- with(
adult,
!is.na(diabetes_dx) & !is.na(age_c) & !is.na(bmi_c) &
!is.na(sex) & !is.na(race)
)
# Subset the survey design using the logical vector (same length as original)
des_cc <- subset(nhanes_design_adult, keep_cc)
# Corresponding complete-case data (optional)
cc <- adult[keep_cc, ] |> droplevels()
cat("\nComplete-case N for survey-weighted model:", nrow(cc), "\n")
Complete-case N for survey-weighted model: 5349 print(table(cc$race))
NH White Mexican American Other Hispanic NH Black
2293 713 470 1101
Other/Multi
772 print(table(cc$diabetes_dx))
0 1
4752 597 print(table(cc$sex))
Male Female
2551 2798 form_cc <- diabetes_dx ~ age_c + bmi_c + sex + race
svy_fit <- survey::svyglm(formula = form_cc, design = des_cc, family = quasibinomial())
summary(svy_fit)
Call:
svyglm(formula = form_cc, design = des_cc, family = quasibinomial())
Survey design:
subset(nhanes_design_adult, keep_cc)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.67143 0.11935 -22.383 1.68e-08 ***
age_c 1.10833 0.05042 21.981 1.94e-08 ***
bmi_c 0.63412 0.05713 11.099 3.88e-06 ***
sexFemale -0.63844 0.10926 -5.843 0.000386 ***
raceMexican American 0.71091 0.13681 5.196 0.000826 ***
raceOther Hispanic 0.46469 0.13474 3.449 0.008712 **
raceNH Black 0.51221 0.15754 3.251 0.011677 *
raceOther/Multi 0.84460 0.17756 4.757 0.001433 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for quasibinomial family taken to be 0.8455444)
Number of Fisher Scoring iterations: 6## plot(residuals(svy_fit, type='deviance'))
# Survey-weighted OR table (no intercept)
svy_or <- broom::tidy(svy_fit, conf.int = TRUE) %>%
dplyr::mutate(OR = exp(estimate), LCL = exp(conf.low), UCL = exp(conf.high)) %>%
dplyr::select(term, OR, LCL, UCL, p.value) %>%
dplyr::filter(term != "(Intercept)")
knitr::kable(svy_or, caption = "Survey-weighted odds ratios (per 1 SD)")| term | OR | LCL | UCL | p.value |
|---|---|---|---|---|
| age_c | 3.0292807 | 2.6967690 | 3.4027912 | 0.0000000 |
| bmi_c | 1.8853571 | 1.6526296 | 2.1508579 | 0.0000039 |
| sexFemale | 0.5281132 | 0.4104905 | 0.6794397 | 0.0003857 |
| raceMexican American | 2.0358434 | 1.4850041 | 2.7910081 | 0.0008262 |
| raceOther Hispanic | 1.5915182 | 1.1664529 | 2.1714810 | 0.0087119 |
| raceNH Black | 1.6689718 | 1.1605895 | 2.4000450 | 0.0116773 |
| raceOther/Multi | 2.3270527 | 1.5451752 | 3.5045697 | 0.0014331 |
The Survey-weighted odds ratios (per 1 SD) are presented. The residual plot looks okay and does not show any pattern.
MICE performed on the imputed dataset - model summary of the pooled imputed dataset presents odds ratio.
# ----- Multiple Imputation (predictors only)
mi_dat <- adult %>%
dplyr::select(diabetes_dx, age, bmi, sex, race, WTMEC2YR, SDMVPSU, SDMVSTRA)
meth <- mice::make.method(mi_dat)
pred <- mice::make.predictorMatrix(mi_dat)
# Do not impute outcome
meth["diabetes_dx"] <- ""
pred["diabetes_dx", ] <- 0
pred[,"diabetes_dx"] <- 1
# Imputation methods
meth["age"] <- "norm"
meth["bmi"] <- "pmm"
meth["sex"] <- "polyreg"
meth["race"] <- "polyreg"
# Survey design vars as auxiliaries only
meth[c("WTMEC2YR","SDMVPSU","SDMVSTRA")] <- ""
pred[, c("WTMEC2YR","SDMVPSU","SDMVSTRA")] <- 1
glimpse(mi_dat)Rows: 5,769
Columns: 8
$ diabetes_dx <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ age <dbl> 69, 54, 72, 73, 56, 61, 42, 56, 65, 26, 76, 33, 32, 38, 50…
$ bmi <dbl> 26.7, 28.6, 28.9, 19.7, 41.7, 35.7, NA, 26.5, 22.0, 20.3, …
$ sex <fct> Male, Male, Male, Female, Male, Female, Male, Female, Male…
$ race <fct> NH Black, NH White, NH White, NH White, Mexican American, …
$ WTMEC2YR <dbl> 13481.04, 24471.77, 57193.29, 65541.87, 25344.99, 61758.65…
$ SDMVPSU <dbl> 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2…
$ SDMVSTRA <dbl> 112, 108, 109, 116, 111, 114, 106, 112, 112, 113, 116, 114…imp <- mice::mice(mi_dat, m = 5, method = meth, predictorMatrix = pred, seed = 123)
iter imp variable
1 1 bmi
1 2 bmi
1 3 bmi
1 4 bmi
1 5 bmi
2 1 bmi
2 2 bmi
2 3 bmi
2 4 bmi
2 5 bmi
3 1 bmi
3 2 bmi
3 3 bmi
3 4 bmi
3 5 bmi
4 1 bmi
4 2 bmi
4 3 bmi
4 4 bmi
4 5 bmi
5 1 bmi
5 2 bmi
5 3 bmi
5 4 bmi
5 5 bmifit_mi <- with(imp, {
age_c <- as.numeric(scale(age))
bmi_c <- as.numeric(scale(bmi))
glm(diabetes_dx ~ age_c + bmi_c + sex + race, family = binomial())
})
pool_mi <- pool(fit_mi)
summary(pool_mi) term estimate std.error statistic df p.value
1 (Intercept) -2.6895645 0.09941301 -27.054453 5566.204 1.486581e-151
2 age_c 1.0660265 0.05594733 19.054108 5520.446 1.911564e-78
3 bmi_c 0.5468538 0.04473386 12.224604 5148.557 6.751227e-34
4 sexFemale -0.6178297 0.09379129 -6.587282 5551.660 4.892566e-11
5 raceMexican American 0.8877355 0.13750463 6.456041 5472.583 1.167455e-10
6 raceOther Hispanic 0.5606621 0.17485537 3.206433 5573.987 1.351505e-03
7 raceNH Black 0.6809629 0.11981185 5.683602 5576.734 1.385727e-08
8 raceOther/Multi 0.7476406 0.15300663 4.886328 4749.963 1.061140e-06## table
mi_or <- summary(pool_mi, conf.int = TRUE, exponentiate = TRUE) %>%
dplyr::rename(
term = term, OR = estimate, LCL = `2.5 %`, UCL = `97.5 %`, p.value = p.value
) %>%
dplyr::filter(term != "(Intercept)")
knitr::kable(mi_or, caption = "MI pooled odds ratios (per 1 SD)")| term | OR | std.error | statistic | df | p.value | LCL | UCL | conf.low | conf.high | |
|---|---|---|---|---|---|---|---|---|---|---|
| 2 | age_c | 2.9038183 | 0.0559473 | 19.054108 | 5520.446 | 0.0000000 | 2.6021752 | 3.2404277 | 2.6021752 | 3.2404277 |
| 3 | bmi_c | 1.7278084 | 0.0447339 | 12.224604 | 5148.557 | 0.0000000 | 1.5827382 | 1.8861754 | 1.5827382 | 1.8861754 |
| 4 | sexFemale | 0.5391132 | 0.0937913 | -6.587282 | 5551.660 | 0.0000000 | 0.4485669 | 0.6479368 | 0.4485669 | 0.6479368 |
| 5 | raceMexican American | 2.4296216 | 0.1375046 | 6.456041 | 5472.583 | 0.0000000 | 1.8555327 | 3.1813298 | 1.8555327 | 3.1813298 |
| 6 | raceOther Hispanic | 1.7518320 | 0.1748554 | 3.206433 | 5573.987 | 0.0013515 | 1.2434346 | 2.4680953 | 1.2434346 | 2.4680953 |
| 7 | raceNH Black | 1.9757793 | 0.1198118 | 5.683602 | 5576.734 | 0.0000000 | 1.5621842 | 2.4988753 | 1.5621842 | 2.4988753 |
| 8 | raceOther/Multi | 2.1120110 | 0.1530066 | 4.886328 | 4749.963 | 0.0000011 | 1.5646727 | 2.8508138 | 1.5646727 | 2.8508138 |
library(gt)
# Bayesian Logistic Regression (formula weights)
adult_imp1 <- complete(imp, 1) %>%
dplyr::mutate(
age_c = as.numeric(scale(age)),
bmi_c = as.numeric(scale(bmi)),
wt_norm = WTMEC2YR / mean(WTMEC2YR, na.rm = TRUE),
# ensure factor refs match survey/mice:
race = forcats::fct_relevel(race, "NH White"),
sex = forcats::fct_relevel(sex, "Male")
) %>%
dplyr::filter(!is.na(diabetes_dx), !is.na(age_c), !is.na(bmi_c),
!is.na(sex), !is.na(race)) %>%
droplevels()
stopifnot(all(is.finite(adult_imp1$wt_norm)))
glimpse(adult_imp1)Rows: 5,592
Columns: 11
$ diabetes_dx <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ age <dbl> 69, 54, 72, 73, 56, 61, 42, 56, 65, 26, 76, 33, 32, 38, 50…
$ bmi <dbl> 26.7, 28.6, 28.9, 19.7, 41.7, 35.7, 23.6, 26.5, 22.0, 20.3…
$ sex <fct> Male, Male, Male, Female, Male, Female, Male, Female, Male…
$ race <fct> NH Black, NH White, NH White, NH White, Mexican American, …
$ WTMEC2YR <dbl> 13481.04, 24471.77, 57193.29, 65541.87, 25344.99, 61758.65…
$ SDMVPSU <dbl> 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2…
$ SDMVSTRA <dbl> 112, 108, 109, 116, 111, 114, 106, 112, 112, 113, 116, 114…
$ age_c <dbl> 1.13241831, 0.27835981, 1.30323001, 1.36016725, 0.39223428…
$ bmi_c <dbl> -0.33319172, -0.06755778, -0.02561558, -1.31184309, 1.7639…
$ wt_norm <dbl> 0.3393916, 0.6160884, 1.4398681, 1.6500477, 0.6380722, 1.5…priors <- c(
set_prior("normal(0, 2.5)", class = "b"),
set_prior("student_t(3, 0, 10)", class = "Intercept")
)
bayes_fit <- brm(
formula = diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race,
data = adult_imp1,
family = bernoulli(link = "logit"),
prior = priors,
chains = 4, iter = 2000, seed = 123,
control = list(adapt_delta = 0.95),
refresh = 0 # quiet Stan output
)Running /opt/R/4.4.2/lib/R/bin/R CMD SHLIB foo.c
using C compiler: ‘gcc (GCC) 11.5.0 20240719 (Red Hat 11.5.0-2)’
gcc -I"/opt/R/4.4.2/lib/R/include" -DNDEBUG -I"/opt/R/4.4.2/lib/R/library/Rcpp/include/" -I"/opt/R/4.4.2/lib/R/library/RcppEigen/include/" -I"/opt/R/4.4.2/lib/R/library/RcppEigen/include/unsupported" -I"/opt/R/4.4.2/lib/R/library/BH/include" -I"/opt/R/4.4.2/lib/R/library/StanHeaders/include/src/" -I"/opt/R/4.4.2/lib/R/library/StanHeaders/include/" -I"/opt/R/4.4.2/lib/R/library/RcppParallel/include/" -I"/opt/R/4.4.2/lib/R/library/rstan/include" -DEIGEN_NO_DEBUG -DBOOST_DISABLE_ASSERTS -DBOOST_PENDING_INTEGER_LOG2_HPP -DSTAN_THREADS -DUSE_STANC3 -DSTRICT_R_HEADERS -DBOOST_PHOENIX_NO_VARIADIC_EXPRESSION -D_HAS_AUTO_PTR_ETC=0 -include '/opt/R/4.4.2/lib/R/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp' -D_REENTRANT -DRCPP_PARALLEL_USE_TBB=1 -I/usr/local/include -fpic -g -O2 -c foo.c -o foo.o
In file included from /opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/Core:19,
from /opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/Dense:1,
from /opt/R/4.4.2/lib/R/library/StanHeaders/include/stan/math/prim/fun/Eigen.hpp:22,
from <command-line>:
/opt/R/4.4.2/lib/R/library/RcppEigen/include/Eigen/src/Core/util/Macros.h:679:10: fatal error: cmath: No such file or directory
679 | #include <cmath>
| ^~~~~~~
compilation terminated.
make: *** [/opt/R/4.4.2/lib/R/etc/Makeconf:195: foo.o] Error 1prior_summary(bayes_fit) prior class coef group resp dpar nlpar lb ub
normal(0, 2.5) b
normal(0, 2.5) b age_c
normal(0, 2.5) b bmi_c
normal(0, 2.5) b raceMexicanAmerican
normal(0, 2.5) b raceNHBlack
normal(0, 2.5) b raceOtherDMulti
normal(0, 2.5) b raceOtherHispanic
normal(0, 2.5) b sexFemale
student_t(3, 0, 10) Intercept
tag source
user
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
usersummary(bayes_fit) # Bayesian model summary Family: bernoulli
Links: mu = logit
Formula: diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race
Data: adult_imp1 (Number of observations: 5592)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -2.66 0.09 -2.84 -2.50 1.00 4187 3510
age_c 1.09 0.06 0.97 1.22 1.00 3012 3098
bmi_c 0.63 0.05 0.53 0.72 1.00 3472 3315
sexFemale -0.66 0.10 -0.86 -0.46 1.00 4003 3052
raceMexicanAmerican 0.69 0.18 0.35 1.04 1.00 3526 2843
raceOtherHispanic 0.43 0.24 -0.07 0.89 1.00 4058 3114
raceNHBlack 0.54 0.15 0.24 0.82 1.00 3597 3177
raceOtherDMulti 0.82 0.19 0.45 1.19 1.00 3763 3257
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).Bayesian Logistic Regression is performed on the imputed dataset from the pooled imputed adult dataset (adult_imp1) and provided the Bayesian model summary statistics, visualization, summary tables.
# Class distribution
ggplot(adult_imp1, aes(x = factor(diabetes_dx))) +
geom_bar(fill = "steelblue") +
labs(
title = "Diabetes Diagnosis Distribution",
x = "Diabetes Diagnosis (0 = No, 1 = Yes)",
y = "Count"
) +
theme_minimal()
prop.table(table(adult_imp1$diabetes_dx))
0 1
0.889485 0.110515 # Summaries of numeric predictors
numeric_vars <- sapply(adult_imp1, is.numeric)
summary(adult_imp1[, numeric_vars]) diabetes_dx age bmi WTMEC2YR
Min. :0.0000 Min. :20.00 Min. :14.1 Min. : 0
1st Qu.:0.0000 1st Qu.:34.00 1st Qu.:24.1 1st Qu.: 18785
Median :0.0000 Median :48.00 Median :27.7 Median : 27803
Mean :0.1105 Mean :48.84 Mean :29.0 Mean : 39779
3rd Qu.:0.0000 3rd Qu.:63.00 3rd Qu.:32.4 3rd Qu.: 52293
Max. :1.0000 Max. :80.00 Max. :82.9 Max. :171395
SDMVPSU SDMVSTRA age_c bmi_c
Min. :1.000 Min. :104.0 Min. :-1.65751 Min. :-2.09476
1st Qu.:1.000 1st Qu.:107.0 1st Qu.:-0.86038 1st Qu.:-0.69669
Median :1.000 Median :111.0 Median :-0.06326 Median :-0.19338
Mean :1.486 Mean :110.9 Mean :-0.01563 Mean :-0.01133
3rd Qu.:2.000 3rd Qu.:115.0 3rd Qu.: 0.79079 3rd Qu.: 0.46371
Max. :2.000 Max. :118.0 Max. : 1.75873 Max. : 7.52398
wt_norm
Min. :0.0000
1st Qu.:0.4729
Median :0.7000
Mean :1.0014
3rd Qu.:1.3165
Max. :4.3150 # Summaries of factor predictors
factor_vars <- sapply(adult_imp1, is.factor)
summary(adult_imp1[, factor_vars]) sex race
Male :2669 NH White :2398
Female:2923 Mexican American: 742
Other Hispanic : 489
NH Black :1141
Other/Multi : 822 # Save your dataset as CSV
write.csv(adult_imp1, "adult_imp1.csv", row.names = FALSE)To evaluate how well the model learned patterns from the training dataset, and how reliably it will perform on new data in real-world scenarios, we decided to perform Training/Testing on the imputed data
Testing set to evaluate model performance on unseen data. - Model Accuracy / Performance - Generalizability - Helps check for overfitting or underfitting - Reliability of Predictions - Confidence on model predictions for new individuals - Distribution and Bias Check - to ensure the training and testing sets are representative of the same population
Results from training and test data- Training (n=4473) & Testing (n=1119) sets: Age ~48 yrs, BMI ~28, bmi_c ~0, wt_norm ~1; ranges similar across sets.
#Code here
import pandas as pd
# Load the data exported from R
adult_data_py = pd.read_csv("adult_imp1.csv") # your R dataset saved as CSV
# Now you can rename it freely in Python
df = adult_data_py # just create a new variable pointing to the same DataFrame
# Check the data
df.head() diabetes_dx age bmi sex ... SDMVSTRA age_c bmi_c wt_norm
0 1 69 26.7 Male ... 112 1.132418 -0.333192 0.339392
1 1 54 28.6 Male ... 108 0.278360 -0.067558 0.616088
2 1 72 28.9 Male ... 109 1.303230 -0.025616 1.439868
3 0 73 19.7 Female ... 116 1.360167 -1.311843 1.650048
4 0 56 41.7 Male ... 111 0.392234 1.763918 0.638072
[5 rows x 11 columns]import pandas as pd
from sklearn.model_selection import train_test_split
# Optional: rename columns if needed
df = df.rename(columns={"diabetes_dx": "diabetes_status"})
# Separate predictors (X) and outcome (y)
X = df.drop(columns=["diabetes_status"])
y = df["diabetes_status"]
# Create 80/20 train-test split, stratifying by outcome to keep class balance
X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=0.2, random_state=1234, stratify=y
)
# Combine X and y if you want full train/test datasets
train_data = pd.concat([X_train, y_train], axis=1)
test_data = pd.concat([X_test, y_test], axis=1)
# Check sizes
print("Training set:", train_data.shape)Training set: (4473, 11)print("Testing set: ", test_data.shape)Testing set: (1119, 11)# Check class balance
print("\nTraining set class distribution:\n", y_train.value_counts(normalize=True))
Training set class distribution:
diabetes_status
0 0.88956
1 0.11044
Name: proportion, dtype: float64print("\nTesting set class distribution:\n", y_test.value_counts(normalize=True))
Testing set class distribution:
diabetes_status
0 0.889187
1 0.110813
Name: proportion, dtype: float64import matplotlib.pyplot as plt
import seaborn as sns
import pandas as pd
# Combine train and test distributions into a DataFrame
train_dist = pd.DataFrame({'Class': y_train.unique(),
'Proportion': y_train.value_counts(normalize=True).values,
'Set': 'Train'})
test_dist = pd.DataFrame({'Class': y_test.unique(),
'Proportion': y_test.value_counts(normalize=True).values,
'Set': 'Test'})
dist_df = pd.concat([train_dist, test_dist])
# Plot
plt.figure(figsize=(8,5))
sns.barplot(x='Class', y='Proportion', hue='Set', data=dist_df)
plt.title('Class Distribution in Training and Testing Sets')
plt.ylabel('Proportion')
plt.xlabel('Diabetes Status')
plt.ylim(0,1)(0.0, 1.0)plt.show()
# Numeric summaries for predictors
print("Training set numeric summary:\n", X_train.describe())Training set numeric summary:
age bmi ... bmi_c wt_norm
count 4473.000000 4473.000000 ... 4473.000000 4473.000000
mean 48.926895 28.933736 ... -0.020899 1.001234
std 17.571583 6.935321 ... 0.969609 0.786057
min 20.000000 14.100000 ... -2.094764 0.000000
25% 34.000000 24.100000 ... -0.696691 0.477152
50% 48.000000 27.700000 ... -0.193384 0.700259
75% 63.000000 32.300000 ... 0.449729 1.286732
max 80.000000 77.500000 ... 6.769021 4.314957
[8 rows x 8 columns]print("\nTesting set numeric summary:\n", X_test.describe())
Testing set numeric summary:
age bmi ... bmi_c wt_norm
count 1119.000000 1119.000000 ... 1119.000000 1119.000000
mean 48.475424 29.275782 ... 0.026921 1.002283
std 17.568193 7.744067 ... 1.082677 0.785208
min 20.000000 15.200000 ... -1.940976 0.000000
25% 33.000000 23.900000 ... -0.724652 0.453099
50% 47.000000 27.800000 ... -0.179404 0.697037
75% 63.000000 32.900000 ... 0.533614 1.378142
max 80.000000 82.900000 ... 7.523981 3.877294
[8 rows x 8 columns]Training (n=4473) & Testing (n=1119) sets: Age ~48 yrs, BMI ~28, bmi_c ~0, wt_norm ~1; ranges similar across sets.
Prior - We used student_t(3, 0, 10) prior R. V. D. Schoot et al. (2013) - student_t(ν,μ,σ)
Student’s t-distribution with: ν=3: degrees of freedom (controls tail heaviness) μ=0: location (center, like the mean) σ=10: scale (spread, like standard deviation) student_t(3, 0, 10) means: It’s centered at 0 It allows large values (± several times 10) It has heavy tails, since ν=3, allows for outliers or unexpected large parameter values Helps the model remain stable, especially with: Small datasets High correlation between predictors Potential outliers in the data
Bayesian Logistic Regression Analysis
###
library(ggplot2)
# priors for two coefficients (age and bmi) prior = N(0,2.5)
prior_draws <- tibble(
term = rep(c("Age (per 1 SD)", "BMI (per 1 SD)"), each = 4000),
value = c(rnorm(4000, 0, 2.5), rnorm(4000, 0, 2.5))
)
ggplot(prior_draws, aes(x = value, fill = term)) +
geom_density(alpha = 0.5) +
theme_minimal() +
labs(title = "Prior Distributions for Coefficients",
x = "Coefficient Value", y = "Density") +
scale_fill_manual(values = c("skyblue", "orange"))
# Diabetes vs BMI
library(ggplot2)
# Create the plot
ggplot(adult_imp1, aes(x = factor(diabetes_dx), y = bmi, fill = factor(diabetes_dx))) +
geom_boxplot(alpha = 0.7) +
scale_x_discrete(labels = c("0" = "No Diabetes", "1" = "Diabetes")) +
labs(
x = "Diabetes Diagnosis",
y = "BMI",
title = "BMI Distribution by Diabetes Status"
) +
theme_minimal() +
theme(legend.position = "none")
# logistic regression curve
ggplot(adult_imp1, aes(x = bmi, y = diabetes_dx)) +
geom_point(aes(y = diabetes_dx), alpha = 0.2, position = position_jitter(height = 0.02)) +
geom_smooth(method = "glm", method.args = list(family = "binomial"), se = TRUE, color = "blue") +
labs(
x = "BMI",
y = "Probability of Diabetes",
title = "Predicted Probability of Diabetes vs BMI"
) +
theme_minimal()
Once we get posterior draws, we calculated summary stats Mean, median, 95% credible intervals summary(bayes_fit) or posterior_summary(bayes_fit) -
summary(bayes_fit) Family: bernoulli
Links: mu = logit
Formula: diabetes_dx | weights(wt_norm) ~ age_c + bmi_c + sex + race
Data: adult_imp1 (Number of observations: 5592)
Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 4000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -2.66 0.09 -2.84 -2.50 1.00 4187 3510
age_c 1.09 0.06 0.97 1.22 1.00 3012 3098
bmi_c 0.63 0.05 0.53 0.72 1.00 3472 3315
sexFemale -0.66 0.10 -0.86 -0.46 1.00 4003 3052
raceMexicanAmerican 0.69 0.18 0.35 1.04 1.00 3526 2843
raceOtherHispanic 0.43 0.24 -0.07 0.89 1.00 4058 3114
raceNHBlack 0.54 0.15 0.24 0.82 1.00 3597 3177
raceOtherDMulti 0.82 0.19 0.45 1.19 1.00 3763 3257
Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).plot(bayes_fit) # Posterior distributions
pp_check(bayes_fit) # Posterior predictive checks
mcmc_trace(bayes_fit) # Convergence (optional)
bayes_R2(bayes_fit) # Model fit Estimate Est.Error Q2.5 Q97.5
R2 0.1313342 0.01265055 0.1064607 0.156078Below are the reported odds ratio from the posterior predicted values and the Bayesian regression
# Posterior ORs (drop intercept, clean labels)
bayes_or <- posterior_summary(bayes_fit, pars = "^b_") %>%
as.data.frame() %>%
tibble::rownames_to_column("raw") %>%
dplyr::mutate(
term = gsub("^b_", "", raw),
term = gsub("race", "race:", term),
term = gsub("sex", "sex:", term),
term = gsub("OtherDMulti", "Other/Multi", term),
term = gsub("OtherHispanic", "Other Hispanic", term),
OR = exp(Estimate),
LCL = exp(Q2.5),
UCL = exp(Q97.5)
) %>%
dplyr::select(term, OR, LCL, UCL) %>%
dplyr::filter(term != "Intercept")
knitr::kable(
bayes_or %>%
dplyr::mutate(dplyr::across(c(OR,LCL,UCL), ~round(.x, 2))),
digits = 2,
caption = "Bayesian posterior odds ratios (95% CrI) — reference: NH White (race), Male (sex)"
)| term | OR | LCL | UCL |
|---|---|---|---|
| age_c | 2.99 | 2.64 | 3.37 |
| bmi_c | 1.87 | 1.71 | 2.05 |
| sex:Female | 0.52 | 0.42 | 0.63 |
| race:MexicanAmerican | 2.00 | 1.41 | 2.84 |
| race:Other Hispanic | 1.54 | 0.93 | 2.43 |
| race:NHBlack | 1.71 | 1.28 | 2.27 |
| race:Other/Multi | 2.27 | 1.56 | 3.28 |
# Combined table
if (!dir.exists("outputs")) dir.create("outputs", recursive = TRUE)
saveRDS(svy_fit, "outputs/svy_fit.rds")
saveRDS(pool_mi, "outputs/pool_mi.rds")
saveRDS(bayes_fit, "outputs/bayes_fit.rds")
saveRDS(svy_or, "outputs/survey_OR_table.rds")
saveRDS(mi_or, "outputs/mi_OR_table.rds")
saveRDS(bayes_or, "outputs/bayes_OR_table.rds")# Results
#Build compact results table (BMI & Age only)
library(dplyr);
library(tidyr);
library(knitr);
library(stringr)
# pretty "OR (LCL–UCL)" string
fmt_or <- function(or, lcl, ucl, digits = 2) {
paste0(
formatC(or, format = "f", digits = digits), " (",
formatC(lcl, format = "f", digits = digits), "–",
formatC(ucl, format = "f", digits = digits), ")"
)
}
# guardrails: require these to exist from Modeling
stopifnot(exists("svy_or"), exists("mi_or"), exists("bayes_or"))
for (nm in c("svy_or","mi_or","bayes_or")) {
if (!all(c("term","OR","LCL","UCL") %in% names(get(nm)))) {
stop(nm, " must have columns: term, OR, LCL, UCL")
}
}
svy_tbl <- svy_or %>% mutate(Model = "Survey-weighted MLE")
mi_tbl <- mi_or %>% mutate(Model = "mice pooled")
bayes_tbl <- bayes_or %>% mutate(Model = "Bayesian")
all_tbl <- bind_rows(svy_tbl, mi_tbl, bayes_tbl) %>%
mutate(term = case_when(
str_detect(term, "bmi_c|\\bBMI\\b") ~ "BMI (per 1 SD)",
str_detect(term, "age_c|\\bAge\\b") ~ "Age (per 1 SD)",
TRUE ~ term
)) %>%
filter(term %in% c("BMI (per 1 SD)", "Age (per 1 SD)")) %>%
mutate(OR_CI = fmt_or(OR, LCL, UCL, digits = 2)) %>%
select(Model, term, OR_CI) %>%
arrange(
factor(Model, levels = c("Survey-weighted MLE","mice pooled","Bayesian")),
factor(term, levels = c("BMI (per 1 SD)","Age (per 1 SD)"))
)
res_wide <- all_tbl %>%
pivot_wider(names_from = term, values_from = OR_CI) %>%
rename(
`BMI (per 1 SD) OR (95% CI)` = `BMI (per 1 SD)`,
`Age (per 1 SD) OR (95% CI)` = `Age (per 1 SD)`
)
kable(
res_wide,
align = c("l","c","c"),
caption = "Odds ratios (per 1 SD) with 95% CIs across models"
)| Model | BMI (per 1 SD) OR (95% CI) | Age (per 1 SD) OR (95% CI) |
|---|---|---|
| Survey-weighted MLE | 1.89 (1.65–2.15) | 3.03 (2.70–3.40) |
| mice pooled | 1.73 (1.58–1.89) | 2.90 (2.60–3.24) |
| Bayesian | 1.87 (1.71–2.05) | 2.99 (2.64–3.37) |
To check posterior draws, we present plot through plots as below - pp check, ppc dens plots, ppc (mean and SD) and ppc bars
# Posterior predictive draws
#Posterior predictive checks (binary outcome)
pp_samples <- posterior_predict(bayes_fit, ndraws = 500) # 500 draws
# Check dimensions
dim(pp_samples) # rows = draws, cols = observations[1] 500 5592# Plot overlay of observed vs predicted counts (duplicate image)
ppc_dens_overlay(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:50, ]) +
labs(title = "Posterior Predictive Check: Density Overlay") +
theme_minimal()
ppc_bars(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:50, ])
#PP check for proportions (useful for binary) # mean comparison
## to check if the simulated means match the observed mean
## mean
ppc_stat(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:100, ], stat = "mean") +
labs(title = "Posterior Predictive Check: Mean of Replicates") +
theme_minimal()
## sd
ppc_stat(y = adult_imp1$diabetes_dx, yrep = pp_samples[1:100, ], stat = "sd") +
labs(title = "PPC: Standard Deviation of Replicates") +
theme_minimal()
# Combine observed and predicted into one data frame
plot_data <- adult_imp1 %>%
mutate(
predicted_bmi = predicted[, "Estimate"],
lower_ci = predicted[, "Q2.5"],
upper_ci = predicted[, "Q97.5"],
obs_index = 1:nrow(adult_imp1) # index for x-axis
)
# Line plot
ggplot(plot_data, aes(x = obs_index)) +
geom_line(aes(y = bmi, color = "Observed")) + # observed BMI
geom_line(aes(y = predicted_bmi, color = "Predicted")) + # predicted BMI
geom_ribbon(aes(ymin = lower_ci, ymax = upper_ci), alpha = 0.2) + # uncertainty
labs(x = "Observation", y = "BMI", color = "Legend") +
theme_minimal()
summary(adult_imp1$bmi) Min. 1st Qu. Median Mean 3rd Qu. Max.
14.1 24.1 27.7 29.0 32.4 82.9 summary(plot_data$bmi_c) Min. 1st Qu. Median Mean 3rd Qu. Max.
-2.09476 -0.69669 -0.19338 -0.01133 0.46371 7.52398 Centering for bmi - Summary of original bmi (observed data) and centered version of BMI. - Centering doesn’t change the distribution shape, only shifts it so the mean is zero. - Centering is useful in regression/Bayesian models to improve numerical stability and interpretability of intercepts - Plots showing predicted values of bmi and age (prior vs predicted) and the proportion of diabetes=1 for each draw
prior_summary(bayes_fit) prior class coef group resp dpar nlpar lb ub
normal(0, 2.5) b
normal(0, 2.5) b age_c
normal(0, 2.5) b bmi_c
normal(0, 2.5) b raceMexicanAmerican
normal(0, 2.5) b raceNHBlack
normal(0, 2.5) b raceOtherDMulti
normal(0, 2.5) b raceOtherHispanic
normal(0, 2.5) b sexFemale
student_t(3, 0, 10) Intercept
tag source
user
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
(vectorized)
userprior_draws <- tibble(
term = rep(c("BMI (per 1 SD)", "Age (per 1 SD)"), each = 4000),
estimate = c(rnorm(4000, 0, 1), rnorm(4000, 0, 1)),
type = "Prior"
)
summary(prior_draws) term estimate type
Length:8000 Min. :-3.585220 Length:8000
Class :character 1st Qu.:-0.697054 Class :character
Mode :character Median : 0.000046 Mode :character
Mean :-0.007350
3rd Qu.: 0.668561
Max. : 3.922817 post# A draws_df: 1000 iterations, 4 chains, and 11 variables
b_Intercept b_age_c b_bmi_c b_sexFemale b_raceMexicanAmerican
1 -2.6 1.1 0.70 -0.71 0.67
2 -2.7 1.0 0.62 -0.57 0.65
3 -2.6 1.1 0.65 -0.76 0.63
4 -2.7 1.0 0.65 -0.67 0.82
5 -2.6 1.1 0.61 -0.73 0.75
6 -2.5 1.0 0.60 -0.77 0.61
7 -2.8 1.1 0.66 -0.66 0.52
8 -2.8 1.2 0.67 -0.57 0.94
9 -2.8 1.1 0.65 -0.52 0.84
10 -2.6 1.1 0.67 -0.85 0.70
b_raceOtherHispanic b_raceNHBlack b_raceOtherDMulti
1 0.605 0.52 0.95
2 0.338 0.45 0.69
3 0.566 0.63 0.54
4 0.453 0.61 0.78
5 0.090 0.50 0.62
6 0.015 0.48 0.60
7 0.736 0.50 0.84
8 0.913 0.57 1.07
9 0.570 0.66 0.81
10 0.467 0.54 0.97
# ... with 3990 more draws, and 3 more variables
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}head(post)# A draws_df: 6 iterations, 1 chains, and 11 variables
b_Intercept b_age_c b_bmi_c b_sexFemale b_raceMexicanAmerican
1 -2.6 1.1 0.70 -0.71 0.67
2 -2.7 1.0 0.62 -0.57 0.65
3 -2.6 1.1 0.65 -0.76 0.63
4 -2.7 1.0 0.65 -0.67 0.82
5 -2.6 1.1 0.61 -0.73 0.75
6 -2.5 1.0 0.60 -0.77 0.61
b_raceOtherHispanic b_raceNHBlack b_raceOtherDMulti
1 0.605 0.52 0.95
2 0.338 0.45 0.69
3 0.566 0.63 0.54
4 0.453 0.61 0.78
5 0.090 0.50 0.62
6 0.015 0.48 0.60
# ... with 3 more variables
# ... hidden reserved variables {'.chain', '.iteration', '.draw'}names(prior_draws)[1] "term" "estimate" "type" library(brms)
library(tidyr)
# Extract posterior draws
post <- as_draws_df(bayes_fit) %>% # bayes_fit = your brms model
select(b_bmi_c, b_age_c) %>% # select your coefficient columns
pivot_longer(
everything(),
names_to = "term",
values_to = "estimate"
) %>%
mutate(
term = case_when(
term == "b_bmi_c" ~ "BMI (per 1 SD)",
term == "b_age_c" ~ "Age (per 1 SD)"
),
type = "Posterior"
)
## visualization of prior and predicted draws
combined_draws <- bind_rows(prior_draws, post)
library(ggplot2)
ggplot(combined_draws, aes(x = estimate, fill = type)) +
geom_density(alpha = 0.4) +
facet_wrap(~ term, scales = "free", ncol = 2) +
theme_minimal(base_size = 13) +
labs(
title = "Prior vs Posterior Distributions",
x = "Coefficient estimate",
y = "Density",
fill = ""
)
#### Compute proportion of diabetes=1 for each draw
pp_proportion <- rowMeans(pp_samples) # proportion of 1's in each posterior draw
# Summary of posterior proportions
summary(pp_proportion) Min. 1st Qu. Median Mean 3rd Qu. Max.
0.09067 0.10533 0.10944 0.10935 0.11320 0.12715 # Optional: visualize the posterior probability distribution
pp_proportion_df <- tibble(proportion = pp_proportion)
ggplot(pp_proportion_df, aes(x = proportion)) +
geom_histogram(binwidth = 0.01, fill = "skyblue", color = "black") +
labs(
title = "Posterior Distribution of Proportion of Diabetes = 1",
x = "Proportion of Diabetes = 1",
y = "Frequency"
) +
theme_minimal()
library(tidyverse)
# Posterior predicted proportion vector
# pp_proportion <- rowMeans(pp_samples) # if not already done
known_prev <- 0.089 # NHANES prevalence
# Posterior summary
posterior_mean <- mean(pp_proportion)
posterior_ci <- quantile(pp_proportion, c(0.025, 0.975)) # 95% credible interval
# Create a data frame for plotting
pp_df <- tibble(proportion = pp_proportion)
# Plot
ggplot(pp_df, aes(x = proportion)) +
geom_histogram(binwidth = 0.005, fill = "skyblue", color = "black") +
geom_vline(xintercept = known_prev, color = "red", linetype = "dashed", size = 1) +
geom_vline(xintercept = posterior_mean, color = "blue", linetype = "solid", size = 1) +
geom_rect(aes(xmin = posterior_ci[1], xmax = posterior_ci[2], ymin = 0, ymax = Inf),
fill = "blue", alpha = 0.1, inherit.aes = FALSE) +
labs(
title = "Posterior Predicted Diabetes Proportion vs NHANES Prevalence",
subtitle = paste0("Red dashed = NHANES prevalence (", known_prev,
"), Blue solid = Posterior mean (", round(posterior_mean,3), ")"),
x = "Proportion of Diabetes = 1",
y = "Frequency"
) +
theme_minimal()
library(dplyr)
# Posterior predicted proportion
posterior_mean <- mean(pp_proportion)
posterior_ci <- quantile(pp_proportion, c(0.025, 0.975)) # 95% credible interval
# NHANES prevalence with SE from survey::svymean
# Suppose you already have:
# svymean(~diabetes_dx, nhanes_design_adult, na.rm = TRUE)
known_prev <- 0.089 # Mean prevalence
known_se <- 0.0048 # Standard error from survey
# Calculate 95% confidence interval
known_ci <- c(
known_prev - 1.96 * known_se,
known_prev + 1.96 * known_se
)
# Print results
data.frame(
Type = c("Posterior Prediction", "NHANES Prevalence"),
Mean = c(posterior_mean, known_prev),
Lower_95 = c(posterior_ci[1], known_ci[1]),
Upper_95 = c(posterior_ci[2], known_ci[2])
) Type Mean Lower_95 Upper_95
2.5% Posterior Prediction 0.1093519 0.09781831 0.1212446
NHANES Prevalence 0.0890000 0.07959200 0.0984080library(ggplot2)
library(dplyr)
# Create a data frame for plotting
ci_df <- data.frame(
Type = c("Posterior Prediction", "NHANES Prevalence"),
Mean = c(0.1096674, 0.089),
Lower_95 = c(0.09772443, 0.079592),
Upper_95 = c(0.1210658, 0.098408)
)
# Plot
ggplot(ci_df, aes(x = Type, y = Mean, color = Type)) +
geom_point(size = 4) +
geom_errorbar(aes(ymin = Lower_95, ymax = Upper_95), width = 0.2) +
ylim(0, max(ci_df$Upper_95) + 0.02) +
labs(
title = "Comparison of Posterior Predicted Diabetes Proportion vs NHANES Prevalence",
y = "Proportion of Diabetes",
x = ""
) +
theme_minimal(base_size = 14) +
theme(legend.position = "none")
Model**:**
`svyglm(formula = form_cc, design = des_cc, family = quasibinomial())`All predictors are significant: age (p < 0.001) and BMI (p < 0.001) show strong positive associations with the outcome, while being female (p = 0.0004) is negatively associated. Other significant associations include raceMexican American (p = 0.0008), raceOther Hispanic (p = 0.0087), raceNH Black (p = 0.0117), and raceOther/Multi (p = 0.0014).
All predictors are statistically significant.
Positive associations: age (p \< 0.001), BMI (p \< 0.001), and all
race categories compared to reference.
Negative association: being female (p \< 0.001)# Bayesian Logistic Regression
Sampling**:** NUTS (4 chains, 2000 iterations each; 1000 warmup,
4000 post-warmup draws)Posterior distribution of all parameters in the model. Density - plot of posterior samples each parameter (e.g., intercept, slope) into a smoothed density curve, showing most of the posterior probability mass lies for best estimates and uncertainty.
Posterior Predictive Distribution - generated from posterior - predictive draws: 𝑦rep∼𝑝(𝑦new∣𝜃)yrep∼p(ynew∣θ) simulate the data given posterior parameter estimates. - Posterior predictive checks (PPC) compare these simulations to real data to assess model fit.
Incorporating two sources of uncertainty: Parameter uncertainty: captured in the posterior distributions Predictive uncertainty: captured in posterior predictive draws
Combining the two provide credible intervals for predictions, not just point predictions and specifies - Given the BMI, the probability of diabetes is likely between 40–55%.
Autocorrelation Interpretation - To show the correlation of a sample with previous samples (lags) in the MCMC chain. Lag indicates the number of steps between samples. Lag 0 is always 1 (perfect correlation with itself). - Each chain (1–4) shown separately for b_age_c (age coefficient) and b_bmi_c (BMI coefficient) presents autocorrelation drops quickly to near zero after lag 1–2 for both coefficients in all chains suggesting good mixing: successive samples are mostly independent after a short lag. - No persistent high autocorrelation indicates MCMC chains are converging well. - Low autocorrelation, as in your plot, is desirable because it means your posterior estimates are reliable and not biased by chain dependence.
Diabetes Predicted proportion vs population prevalence - Posterior predicted proportion is slightly higher than the observed NHANES prevalence (10.97% vs 8.9%). - Intervals overlap slightly, but the posterior tends to overestimate diabetes compared to NHANES.
The difference could be due to: - Model assumptions (logistic link, priors) - Predictor effects (BMI, age, sex, race) - Sample characteristics vs population weighting in NHANES
The results find our model is reasonable but slightly conservative (predicting higher risk) relative to the observed population prevalence.
Across multiple modeling approaches—survey-weighted maximum likelihood, multiple imputation, and Bayesian regression—both age and BMI were consistently strong predictors of diabetes. Eachstandard deviation increase in age nearly tripled the odds of diabetes, while a similar increase in BMI elevated the odds by approximately 1.7–1.9 times. The consistency of these results across models highlights the robustness of the associations and underscores the importance of age and BMI as key risk factors for diabetes in this population.
Effect stability: point estimates in rhe Bayesian model’s closely aligned with those from the frequentist, indicating that prior regularization stabilized the estimates in the presence of modest missingness.
Uncertainty quantification: Bayesian credible intervals of odds ration were slightly narrower yet overlapped the frequentist confidence intervals, suggest comparable inferential precision while offering improved interpretability.
Design considerations: # Survey-weighted MLE (Maximum Likelihood Estimator) - incorporates each observation weighted according to its survey weight. - provide estimates that reflect the population-level parameters, not just the sample- produces population-representative estimates. # Bayesian model with normalized weights- - instead of fully modeling the survey design, it used normalized sampling weights as importance weights - the scaled weights that sum to the sample size approximates the effect of survey weights, but does not fully account for: Stratification, clustering, design-based variance adjustments. - Bayesian inference treats the weighted likelihood as from a regular model, ignoring some survey design features.
The use of multiple imputation allowed for robust analysis despite missing data, increasing power and reducing bias. Comparison of frequentist and Bayesian models demonstrated consistency in significant predictors, while Bayesian approaches provided the advantage of posterior distributions and probabilistic interpretation. The = Across all models, both age and BMI emerged as strong and consistent predictors of diabetes. The consistency across modeling approaches strengthens the validity of these findings Multiple imputation accounted for potential biases due to missing data, and Bayesian modeling provided robust credible intervals that closely matched frequentist estimates. align with previous epidemiological research indicating that increasing age and higher BMI are among the most important determinants of type 2 diabetes risk.Cumulative exposure to metabolic and lifestyle risk factors over time, and the role of excess adiposity and insulin related effects account for diabetes. Survey weighted dataset strenghthens ensuring population representativeness, multiple imputation to handle missing data, and rigorous Bayesian estimation provided high effective sample sizes and R̂ ≈ 1.00 across parameters confirmed excellent model convergence. Bayesian logistic regression provided inference statistically consistent and interpretable achieving the aim of this study. In future research hierarchical model using NHANES cycles or adding variables (lab tests) could assess nonlinear effects of metabolic risk factors.
Translational Perspective from the Bayesian Diabetes Prediction Project. This project further demonstrates the translational potential of Bayesian modeling in clinical decision-making and public health strategy. By using patient-level predictors such as age, BMI, sex, and race to estimate the probability of diabetes, the model moves beyond descriptive statistics toward individualized risk prediction. The translational move lies in converting these probabilistic outputs into actionable thresholds—such as identifying the BMI or age at which the predicted risk of diabetes exceeds a clinically meaningful level (e.g., 30%). Such insights can guide early screening, personalized lifestyle interventions, and targeted prevention programs for populations at higher risk. This approach embodies precision public health—bridging data science and medical decision-making to deliver tailored, evidence-based strategies that can ultimately improve diabetes prevention and management outcomes.
What changes in modifiable predictors would lower diabetes risk?
# Translational Research Implications: - We now use the model to guide prevention or intervention. - Only BMI is a modifiable risk factor here - What must change in BMI, behavior, or lifestyle to achieve a lower risk threshold? In practice, we hold non modifiable predictors as constant (sex, race). Vary modifiable predictors (BMI) until the model predicts the desired probability.
# Below we used the data of first two particiants from the same dataset (Internal validation - as the data is from the same dataset) # Next we used fake data of a new participant (not in the dataset) - For model validation (External validation as the data is out-of-sample) - It means - applying the trained model to a specific individual’s data (new participanst) — using posterior draws to estimate predictive uncertainty (credible interval). - It is to test generalizability — how well the model performs on new, independent data.
# Use the first participant
# using multiple covariates to select someone
participant1_data <- adult[1, ]
# predicted probabilities for patient 1
phat1 <- posterior_linpred(bayes_fit, newdata = participant1_data, transform = TRUE)
# 'transform = TRUE' gives probabilities for logistic regression
# Store in a data frame for plotting
post_pred_df <- data.frame(pred = phat1)
# Compute 95% credible interval
ci_95_participant1 <- quantile(phat1, c(0.025, 0.975))
# Plot
ggplot(post_pred_df, aes(x = pred)) +
geom_density(color='darkblue', fill='lightblue') +
geom_vline(xintercept = ci_95_participant1[1], color='red', linetype='dashed') +
geom_vline(xintercept = ci_95_participant1[2], color='red', linetype='dashed') +
xlab('Probability of being diabetic (Outcome=1)') +
ggtitle('Posterior Predictive Distribution 95% Credible Interval') +
theme_bw()
library(ggplot2)
new_participant <- data.frame(
age_c = 40,
bmi_c = 25,
sex = "Female",
race = "Mexican American"
)
# Posterior predicted probabilities
phat_new <- posterior_linpred(bayes_fit, newdata = new_participant, transform = TRUE)
# Convert to numeric vector
phat_vec <- as.numeric(phat_new)
# Check the range to see if all values are similar
range(phat_vec)[1] 1 1# Store in a data frame
post_pred_df_new <- data.frame(pred = phat_vec)
# Compute 95% credible interval from the vector
ci_95_new_participant <- quantile(phat_vec, c(0.025, 0.975))
# Plot
ggplot(post_pred_df_new, aes(x = pred)) +
geom_density(color='darkblue', fill='lightblue', alpha = 0.6) +
geom_vline(xintercept = ci_95_new_participant[1], color='red', linetype='dashed') +
geom_vline(xintercept = ci_95_new_participant[2], color='red', linetype='dashed') +
xlim(0, 1) + # ensures you see the curve even if values are close
xlab('Probability of being diabetic (Outcome=1)') +
ggtitle('Posterior Predictive Distribution (95% Credible Interval)') +
theme_bw()
In this analysis, a grid of body mass index (BMI) values is generated to examine the relationship between BMI and the predicted probability of diabetes while holding other covariates (age, sex, and race) constant. - Using the fitted Bayesian logistic regression model, posterior predictive probabilities computed for each BMI value and the mean of posterior draws estimated the average predicted probability of diabetes across the BMI range. - We then tried to identify the BMI value whose predicted probability was closest to a predefined target probability (here, 0.3). - This enabled interpretation of the model in a clinically meaningful way—specifically, by determining the BMI level associated with a 30% predicted probability of diabetes for a given demographic profile. - We were able to link statistical inference to practical health thresholds relevant for risk communication and prevention strategies in translational research.
# Grid of possible BMI values (centered if model used bmi_c)
bmi_seq <- seq(18, 40, by = 0.5)
newdata_grid <- data.frame(
age_c = 40,
bmi_c = bmi_seq,
sex = "Female",
race = "Mexican American"
)
# Posterior mean predicted probabilities
pred_probs <- posterior_linpred(bayes_fit, newdata = newdata_grid, transform = TRUE)
# Average over posterior draws to get the mean predicted probability per BMI
prob_mean <- colMeans(pred_probs)
# Combine with BMI values
pred_df <- cbind(newdata_grid, prob_mean)
target_prob <- 0.3
closest <- pred_df[which.min(abs(pred_df$prob_mean - target_prob)), ]
closest age_c bmi_c sex race prob_mean
1 40 18 Female Mexican American 1ggplot(pred_df, aes(x = bmi_c, y = prob_mean)) +
geom_line(color = "darkblue", linewidth = 1.2) +
geom_hline(yintercept = target_prob, color = "red", linetype = "dashed") +
geom_vline(xintercept = closest$bmi_c, color = "red", linetype = "dotted") +
annotate("text", x = closest$bmi_c, y = target_prob + 0.05,
label = paste0("Target BMI ≈ ", round(closest$bmi_c, 1)),
color = "red", hjust = -0.1) +
labs(
x = "BMI (centered or raw, depending on model)",
y = "Predicted Probability of Diabetes",
title = "Inverse Prediction: BMI Needed for Target Diabetes Risk"
) +
theme_bw()